🔢Algebraic K-Theory Unit 12 – Merkurjev-Suslin Theorem & Bloch-Kato Conjecture

Key Concepts and Definitions

• Galois cohomology groups $H^n(F, \mu_m^{\otimes n})$ play a central role in the study of Merkurjev-Suslin Theorem and Bloch-Kato Conjecture
  • $F$ represents a field
  • $\mu_m$ denotes the group of $m$-th roots of unity
• Milnor K-groups $K_n^M(F)$ are defined as the quotient of the tensor algebra of the multiplicative group $F^\times$ by the ideal generated by the Steinberg relations
• Symbol maps $h_{n,m}: K_n^M(F)/m \to H^n(F, \mu_m^{\otimes n})$ connect Milnor K-groups to Galois cohomology groups
• Norm residue homomorphism $K_n^M(F)/m \to H^n(F, \mu_m^{\otimes n})$ is a generalization of the Hilbert symbol in class field theory
• Brauer group $Br(F)$ classifies central simple algebras over a field $F$ up to Morita equivalence
• Kummer theory describes the Galois group of the extension obtained by adjoining $m$-th roots of elements in a field containing a primitive $m$-th root of unity
• Étale cohomology is a cohomology theory for schemes that is an analogue of singular cohomology for topological spaces

Historical Context and Development

• The study of Galois cohomology and its connection to algebraic K-theory has its roots in the work of Hilbert, Kummer, and others in the late 19th century
• In the 1970s, Milnor introduced the notion of Milnor K-groups, which provided a new perspective on the relationship between Galois cohomology and algebraic K-theory
• Merkurjev and Suslin proved their eponymous theorem in 1982, establishing the bijectivity of the norm residue homomorphism for $n=2$ and arbitrary $m$
  • This result generalized the classical Hilbert symbol and had significant implications for the study of Brauer groups
• Bloch and Kato independently formulated their conjecture in the 1980s, proposing that the norm residue homomorphism is an isomorphism for all $n$ and $m$
  • The conjecture was motivated by connections to motivic cohomology and the study of special values of L-functions
• Voevodsky, Rost, and others made significant progress towards proving the Bloch-Kato Conjecture in the 1990s and early 2000s
  • Their work introduced new techniques, such as motivic cohomology and the slice filtration, which have had far-reaching consequences in algebraic geometry and algebraic K-theory

Merkurjev-Suslin Theorem Explained

• The Merkurjev-Suslin Theorem states that for any field $F$ and any positive integer $m$, the norm residue homomorphism $K_2^M(F)/m \to H^2(F, \mu_m^{\otimes 2})$ is an isomorphism
  • This result establishes a deep connection between Milnor K-groups and Galois cohomology in degree 2
• The proof of the theorem relies on a careful analysis of the structure of central simple algebras over $F$ and their relationship to the Brauer group $Br(F)$
• A key ingredient in the proof is the use of Galois symbols, which are explicit generators of the Galois cohomology groups $H^2(F, \mu_m^{\otimes 2})$
  • These symbols can be constructed using the norm map from cyclic extensions of $F$
• The Merkurjev-Suslin Theorem has important consequences for the study of quadratic forms and the structure of the Witt ring of a field
  • It allows for the computation of the $m$-torsion of the Brauer group $Br(F)[m]$ in terms of generators and relations
• The theorem also plays a crucial role in the proof of the Milnor Conjecture on quadratic forms, which states that the natural map from the Witt ring of a field to its Milnor K-theory is an isomorphism modulo 2-torsion

Bloch-Kato Conjecture Overview

• The Bloch-Kato Conjecture, also known as the norm residue isomorphism theorem, asserts that the norm residue homomorphism $K_n^M(F)/m \to H^n(F, \mu_m^{\otimes n})$ is an isomorphism for all $n$ and $m$
  • This generalizes the Merkurjev-Suslin Theorem to higher degrees and provides a complete description of the relationship between Milnor K-groups and Galois cohomology
• The conjecture has deep connections to motivic cohomology, as it can be reformulated in terms of the motivic Steenrod algebra and the motivic Adams spectral sequence
• Voevodsky's proof of the Bloch-Kato Conjecture relies on the construction of the motivic Eilenberg-MacLane spaces and the development of the slice filtration in motivic homotopy theory
  • The slice filtration provides a way to analyze the structure of motivic cohomology groups and to relate them to Milnor K-groups
• The proof also makes use of the norm varieties introduced by Rost, which are geometric objects that encode information about the norm map in Galois cohomology
• The Bloch-Kato Conjecture has important implications for the study of special values of L-functions and the structure of the algebraic K-theory of fields
  • It provides a way to relate the étale cohomology of algebraic varieties to their motivic cohomology and to the algebraic K-theory of their function fields

Connections to Algebraic K-Theory

• The Merkurjev-Suslin Theorem and the Bloch-Kato Conjecture establish deep connections between Milnor K-groups, Galois cohomology, and algebraic K-theory
• Algebraic K-theory, introduced by Quillen, is a powerful invariant of rings and schemes that generalizes the classical notion of the Grothendieck group
  • The algebraic K-groups $K_n(R)$ of a ring $R$ are defined using the homotopy groups of certain classifying spaces associated to the category of projective modules over $R$
• The Milnor K-groups $K_n^M(F)$ of a field $F$ can be viewed as a "simplified" version of the algebraic K-groups, capturing essential information about the multiplicative structure of $F$
• The norm residue isomorphism theorem implies that the algebraic K-groups of a field $F$ contain important arithmetic information, as they are closely related to the Galois cohomology groups $H^n(F, \mu_m^{\otimes n})$
• The motivic spectral sequence, which relates motivic cohomology to algebraic K-theory, provides a way to study the structure of algebraic K-groups using the tools of motivic homotopy theory
  • The Bloch-Kato Conjecture can be interpreted as a statement about the convergence of the motivic spectral sequence and the structure of the motivic Steenrod algebra
• The study of algebraic K-theory has important applications to the theory of quadratic forms, the structure of the Brauer group, and the classification of algebraic varieties
  • The Merkurjev-Suslin Theorem and the Bloch-Kato Conjecture provide powerful tools for understanding these connections and for computing algebraic K-groups in specific cases

Proof Techniques and Strategies

• The proofs of the Merkurjev-Suslin Theorem and the Bloch-Kato Conjecture rely on a wide range of techniques from algebraic geometry, algebraic topology, and representation theory
• A key idea in the proof of the Merkurjev-Suslin Theorem is the use of Galois symbols to construct explicit generators of the Galois cohomology groups $H^2(F, \mu_m^{\otimes 2})$
  • These symbols are defined using the norm map from cyclic extensions of the field $F$ and satisfy certain compatibility relations
• The proof also involves a careful analysis of the structure of central simple algebras over $F$ and their relationship to the Brauer group $Br(F)$
  • This requires the use of techniques from noncommutative ring theory and the theory of division algebras
• Voevodsky's proof of the Bloch-Kato Conjecture makes essential use of motivic homotopy theory and the slice filtration
  • The slice filtration provides a way to analyze the structure of motivic cohomology groups and to relate them to Milnor K-groups
  • The proof also relies on the construction of the motivic Eilenberg-MacLane spaces, which are geometric objects that represent motivic cohomology groups
• Rost's norm varieties play a crucial role in the proof of the Bloch-Kato Conjecture, as they provide a way to encode information about the norm map in Galois cohomology
  • The construction of these varieties involves techniques from algebraic geometry and the theory of algebraic groups
• The proofs also make use of results from representation theory, such as the theory of Galois representations and the study of the cohomology of linear algebraic groups
  • These techniques are used to analyze the structure of the Galois cohomology groups and to relate them to other invariants, such as étale cohomology and algebraic K-theory

Applications and Implications

• The Merkurjev-Suslin Theorem and the Bloch-Kato Conjecture have numerous applications in algebraic geometry, number theory, and algebraic K-theory
• In algebraic geometry, the theorems provide a way to compute the Brauer group of a field and to study the structure of central simple algebras
  • This has important consequences for the classification of algebraic varieties and the study of their arithmetic properties
• In number theory, the Bloch-Kato Conjecture has deep connections to the study of special values of L-functions and the Tamagawa number conjecture
  • The conjecture provides a way to relate the étale cohomology of algebraic varieties to their motivic cohomology and to the algebraic K-theory of their function fields
• The theorems also have implications for the study of quadratic forms and the structure of the Witt ring of a field
  • The Merkurjev-Suslin Theorem allows for the computation of the $m$-torsion of the Brauer group $Br(F)[m]$ in terms of generators and relations, which has applications to the classification of quadratic forms
• In algebraic K-theory, the Bloch-Kato Conjecture provides a way to understand the structure of the algebraic K-groups of fields and to relate them to Galois cohomology and motivic cohomology
  • This has led to new insights into the structure of algebraic K-theory and its relationship to other invariants, such as the étale cohomology of algebraic varieties
• The techniques developed in the proofs of these theorems, such as motivic homotopy theory and the slice filtration, have found applications in many other areas of mathematics
  • These include the study of algebraic cycles, the theory of motives, and the classification of algebraic varieties

Challenges and Open Questions

• Despite the significant progress made by the Merkurjev-Suslin Theorem and the Bloch-Kato Conjecture, there remain many open questions and challenges in the study of Galois cohomology and algebraic K-theory
• One important problem is the computation of the algebraic K-groups of fields and the determination of their structure as modules over the Galois group
  • While the Bloch-Kato Conjecture provides a way to relate algebraic K-groups to Galois cohomology, the explicit computation of these groups remains a difficult problem in many cases
• Another challenge is the extension of the Merkurjev-Suslin Theorem and the Bloch-Kato Conjecture to more general settings, such as fields with imperfect residue fields or schemes over arbitrary base fields
  • This requires the development of new techniques in motivic homotopy theory and the study of Galois cohomology in non-classical settings
• The relationship between Galois cohomology and other cohomology theories, such as étale cohomology and crystalline cohomology, is another area of active research
  • Understanding the connections between these theories and their relationship to algebraic K-theory is an important problem with applications to arithmetic geometry and number theory
• The study of the Galois action on algebraic K-groups and its relationship to the Galois action on other invariants, such as the étale fundamental group, is another open question
  • This has connections to the Langlands program and the study of Galois representations
• Finally, the application of the ideas and techniques developed in the study of the Merkurjev-Suslin Theorem and the Bloch-Kato Conjecture to other areas of mathematics, such as mathematical physics and topology, remains an active area of research
  • Understanding the connections between these fields and the role of Galois cohomology and algebraic K-theory in these contexts is an important challenge for future work.